A hopfion is a topological soliton.[1] [2] [3] [4] It is a stable three-dimensional localised configuration of a three-component field with a knotted topological structure. They are the three-dimensional counterparts of 2D skyrmions, which exhibit similar topological properties in 2D. Hopfions are widely studied in many physical systems over the last half century, as summarized here http://hopfion.com
The soliton is mobile and stable: i.e. it is protected from a decay by an energy barrier. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician, Heinz Hopf.
A model that supports hopfions was proposed as follows[1]
The terms of higher-order derivatives are required to stabilize the hopfions.
Stable hopfions were predicted within various physical platforms, including Yang-Mills theory,[5] superconductivity[6][7] and magnetism.[8][9][10][4]
Hopfions have been observed experimentally in chiral colloidal magnetic materials [2], in chiral liquid crystals [11], [12] in Ir/Co/Pt multilayers using X-ray magnetic circular dichroism[13] and in the polarization of free-space monochromatic light.[14][15]
In chiral magnets, the hopfion has been theoretically predicted to occur within the spiral magnetic phase, where it was called a "heliknoton".[16] In recent years, the concept of a "fractional hopfion" has also emerged where not all preimages of magnetisation have a nonzero linking.[17][18]